Solving a parabolic equation

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# Set seed for reproducibility
set.seed(1982) 
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knitr::opts_chunk$set(
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captioner <- function(caption) {
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# Define the function to truncate a number to two decimal places
truncate_to_two <- function(x) {
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# install.packages("INLA",repos=c(getOption("repos"),INLA="https://inla.r-inla-download.org/R/testing"), dep=TRUE)
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)

## Loading required package: Matrix

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library(inlabru)

## Loading required package: fmesher

library(rSPDE)

## This is rSPDE 2.5.1
## - See https://davidbolin.github.io/rSPDE for vignettes and manuals.

library(MetricGraph)

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library(grateful)

library(plotly)

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We want to solve the fractional diffusion equation \[\begin{equation} \label{eq:maineq} \partial_t u+(\kappa^2-\Delta_\Gamma)^{\frac{\alpha}{2}} u=f \text { on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma, \end{equation}\] where \(u\) satisfies the Kirchhoff vertex conditions \[\begin{equation} \label{eq:Kcond} \left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V: \sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\} \end{equation}\]

If \(f=0\), then the solution is given by \[\begin{equation} \label{eq:sol_reprentation} u(s,t) = \displaystyle\sum_{j\in\mathbb{N}}e^{-\lambda^{\frac{\alpha}{2}}_jt}\left(u_0, e_j\right)_{L_2(\Gamma)}e_j(s). \end{equation}\]

# Function to build a tadpole graph and create a mesh
gets_graph_tadpole <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 100)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges = list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$build_mesh(h = h)
  return(graph)
}

Let \(\Gamma_T = (\mathcal{V},\mathcal{E})\) characterize the tadpole graph with \(\mathcal{V}= \{v_1,v_2\}\) and \(\mathcal{E}= \{e_1,e_2\}\) as specified in Figure \(\ref{Interval.Circle.Tadpole}\)c. The left edge \(e_1\) has length 1 and the circular edge \(e_2\) has length 2. As discussed in Subsection \(\ref{subsec:prelim}\), a point on \(e_1\) is parameterized via \(s=\left(e_1, t\right)\) for \(t \in[0,1]\) and a point on \(e_2\) via \(s=\left(e_2, t\right)\) for \(t\in[0,2]\). One can verify that \(-\Delta_\Gamma\) has eigenvalues \(0,\left\{(i \pi / 2)^2\right\}_{i \in \mathbb{N}}\) and \(\left\{(i \pi / 2)^2\right\}_{2 i \in \mathbb{N}}\) with corresponding eigenfunctions \(\phi_0\), \(\left\{\phi_i\right\}_{i \in \mathbb{N}}\), and \(\left\{\psi_i\right\}_{2 i \in \mathbb{N}}\) given by \(\phi_0(s)=1 / \sqrt{3}\) and \[\begin{equation*} \phi_i(s)=C_{\phi, i}\begin{cases} -2 \sin (\frac{i\pi}{2}) \cos (\frac{i \pi t}{2}), & s \in e_1, \\ \sin (i \pi t / 2), & s \in e_2, \end{cases}, \quad \psi_i(s)=\frac{\sqrt{3}}{\sqrt{2}} \begin{cases} (-1)^{i / 2} \cos (\frac{i \pi t}{2}), & s \in e_1, \\ \cos (\frac{i \pi t}{2}), & s \in e_2, \end{cases}, \end{equation*}\] where \(C_{\phi, i}=1\) if \(i\) is even and \(C_{\phi, i}=1 / \sqrt{3}\) otherwise. Moreover, these functions form an orthonormal basis for \(L_2(\Gamma_T)\).

# Function to compute the eigenfunctions 
tadpole.eig <- function(k,graph){
x1 <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[graph$mesh$PtE[,1]==1,2]) 
x2 <- c(0,graph$get_edge_lengths()[2]*graph$mesh$PtE[graph$mesh$PtE[,1]==2,2]) 

if(k==0){ 
  f.e1 <- rep(1,length(x1)) 
  f.e2 <- rep(1,length(x2)) 
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  f = list(phi=f1/sqrt(3)) 
  
} else {
  f.e1 <- -2*sin(pi*k*1/2)*cos(pi*k*x1/2) 
  f.e2 <- sin(pi*k*x2/2)                  
  
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  
  if((k %% 2)==1){ 
    f = list(phi=f1/sqrt(3)) 
  } else { 
    f.e1 <- (-1)^{k/2}*cos(pi*k*x1/2)
    f.e2 <- cos(pi*k*x2/2)
    f2 = c(f.e1[1],f.e2[1],f.e1[-1],f.e2[-1]) 
    f <- list(phi=f1,psi=f2/sqrt(3/2))
  }
}

return(f)
}

Implementation of \(u\)

h <- 0.01
graph <- gets_graph_tadpole(h = h)

## Starting graph creation...

## LongLat is set to FALSE

## Creating edges...

## Setting edge weights...

## Computing bounding box...

## Setting up edges

## Merging close vertices

## Total construction time: 0.15 secs

## Creating and updating vertices...

## Storing the initial graph...

## Computing the relative positions of the edges...

T_final <- 2
time_step <- 0.01
time_seq <- seq(0, T_final, by = time_step)
# Compute the FEM matrices
graph$compute_fem()
G <- graph$mesh$G
C <- graph$mesh$C
I <- Matrix::Diagonal(nrow(C))
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
edge_number <- graph$mesh$VtE[, 1]
pos <- sum(edge_number == 1)+1
order_to_plot <- function(v)return(c(v[1], v[3:pos], v[2], v[(pos+1):length(v)], v[2]))
weights <- graph$mesh$weights
# Initial condition
# U_0 <- 10*exp(-((x-1)^2 + (y)^2))

kappa <- 1
alpha <- 0.5
beta <- alpha/2
L <- kappa^2*C + G
op <- fractional.operators(L, beta, C, scale.factor = kappa^2, m = 1)
Pl <- op$Pl
Pr <- op$Pr
Ci <- op$Ci

# Parameters to construct U_0
N_finite <- 4 # choose an even number
adjusted_N_finite <- N_finite + N_finite/2 + 1
EIGENVAL_ALPHA <- NULL
EIGENFUN <- NULL       
for (j in 0:N_finite) {
    lambda_j_alpha <- (kappa^2 + (j*pi/2)^2)^(alpha/2)
    e_j <- tadpole.eig(j,graph)$phi
    EIGENVAL_ALPHA <- c(EIGENVAL_ALPHA, lambda_j_alpha)         
    EIGENFUN <- cbind(EIGENFUN, e_j)            
    if (j>0 && (j %% 2 == 0)) {
      lambda_j_alpha <- (kappa^2 + (j*pi/2)^2)^(alpha/2)
      e_j <- tadpole.eig(j,graph)$psi
      EIGENVAL_ALPHA <- c(EIGENVAL_ALPHA, lambda_j_alpha)         
      EIGENFUN <- cbind(EIGENFUN, e_j)           
    }
}

# Building the initial condition as \sum coeff_j EIGENFUN_j
coeff <- 1*(1:adjusted_N_finite)^-1
coeff[-5] <- 0
# lower_zeroer <- 10  # choose an even number
# adjusted_lower_zeroer <- lower_zeroer + lower_zeroer/2 + 1
# coeff[adjusted_lower_zeroer:adjusted_N_finite] <- 0
U_0 <- EIGENFUN %*% coeff

# Building the true solution as \sum coeff_j EIGENFUN_j e^{-\lambda_j^{\frac{\alpha}{2}}t}
U_true <- matrix(NA, nrow = length(x), ncol = length(time_seq))
for (k in 1:length(time_seq)) {
  aux_k <- rep(0, length(x))
  for (j in 1:adjusted_N_finite) {
    aux_k <- aux_k + exp(-time_seq[k]*EIGENVAL_ALPHA[j])*coeff[j]*EIGENFUN[, j]
  }
  U_true[, k] <- aux_k
}

{r}
Pr.apply.mult <- function(v){return(Pr.mult(op, v))}
Pl.apply.solve <- function(v){return(Pl.solve(op, v))}
PliC <- apply(C, 2, Pl.apply.solve) # PlC^-1
# Precompute the LHS1 matrix
aux <- apply(PliC, 2, Pr.apply.mult) #PrPl^-1C
LHS <- aux + time_step * Matrix::Diagonal(nrow(C)) 

# Initialize U matrix to store solution at each time step
U_approx <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_approx[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  # Compute the right-hand side for the second equation
  RHS <- aux %*% U_approx[, k]
  U_approx[, k + 1] <- as.matrix(solve(LHS, RHS))
}

LHS <- t(Pr)%*% C + time_step * t(Pl)
# Initialize U matrix to store solution at each time step
U_approx <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_approx[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  # Compute the right-hand side for the second equation
  RHS <- t(Pr)%*% C %*% U_approx[, k]
  U_approx[, k + 1] <- as.matrix(solve(LHS, RHS))
}

{r}
Pr.apply.mult <- function(v){return(Pr.mult(op, v))}
aa <- apply(C, 2, Pr.apply.mult)
LHS <- aa + time_step * Pl
# Initialize U matrix to store solution at each time step
U_approx <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_approx[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  # Compute the right-hand side for the second equation
  RHS <- aa %*% U_approx[, k]
  U_approx[, k + 1] <- as.matrix(solve(LHS, RHS))
}

x <- order_to_plot(x)
y <- order_to_plot(y)
max_error_at_each_time <- apply((U_true - U_approx)^2, 2, mean)

U_true <- apply(U_true, 2, order_to_plot)
U_approx <- apply(U_approx, 2, order_to_plot)



# Create interactive plot
fig <- plot_ly()


# Add second line (max_error_at_each_time)
fig <- fig %>% add_trace(
  x = ~time_seq, y = ~max_error_at_each_time, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'red', width = 2),
  marker = list(size = 4),
  name = "Max Error True and Approx 2"
)

# Layout
fig <- fig %>% layout(
  title = "Max Error at Each Time Step",
  xaxis = list(title = "Time"),
  yaxis = list(title = "Max Error"),
  legend = list(x = 0.1, y = 0.9)
)


plot_data <- data.frame(
  x = rep(x, times = ncol(U_true)),
  y = rep(y, times = ncol(U_true)),
  z_true = as.vector(U_true),
  z_approx = as.vector(U_approx),
  frame = rep(time_seq, each = length(x))
)

# Compute axis limits
x_range <- range(x)
y_range <- range(y)
z_range <- range(c(U_true, U_approx))

# Initial plot setup (first frame only)
p <- plot_ly(plot_data, frame = ~frame) %>%
  add_trace(
    x = ~x, y = ~y, z = ~z_true,
    type = "scatter3d", mode = "lines",
    name = "True",
    line = list(color = "blue", width = 2)
  ) %>%
  add_trace(
    x = ~x, y = ~y, z = ~z_approx,
    type = "scatter3d", mode = "lines",
    name = "Approx",
    line = list(color = "red", width = 2)
  ) %>%
  layout(
    scene = list(
      xaxis = list(title = "x", range = x_range),
      yaxis = list(title = "y", range = y_range),
      zaxis = list(title = "Value", range = z_range),
      aspectratio = list(x = 2.4, y = 1.2, z = 1.2),
           camera = list(
      eye = list(x = 1.5, y = 1.5, z = 1),  # Adjust the viewpoint
      center = list(x = 0, y = 0, z = 0))
    ),
    updatemenus = list(
      list(
        type = "buttons", showactive = FALSE,
        buttons = list(
          list(label = "Play", method = "animate",
               args = list(NULL, list(frame = list(duration = 100, redraw = TRUE), fromcurrent = TRUE))),
          list(label = "Pause", method = "animate",
               args = list(NULL, list(mode = "immediate", frame = list(duration = 0), redraw = FALSE)))
        )
      )
    ),
    title = "Time: 0"
  )

# Convert to plotly object with frame info
pb <- plotly_build(p)

# Inject custom titles into each frame
for (i in seq_along(pb$x$frames)) {
  t <- time_seq[i]
  err <- signif(max_error_at_each_time[i], 4)
  pb$x$frames[[i]]$layout <- list(title = paste0("Time: ", t, " | Max Error: ", err))
}

# Create 3D scatter plot
pic <- plot_ly(x = ~x, y = ~y, z = ~U_true[,2], type = 'scatter3d', mode = 'lines')

fig  # Display the plot

pb

Figure 1: Caption

pic